I am working on proving the theorem that,
For a connected multi-graph G, G is Eulerian if and only if every vertex has even degree. By Eulerian here, I mean a graph that has a Eulerian Circuit. In this question I am focused on the (=>) direction of the proof.
I am using the following Proof, which is as followed:
If G is Eulerian then there is an Euler circuit, P, in G. Every time a vertex is listed, that accounts for two edges adjacent to that vertex, the one before it in the list and the one after it in the list. This circuit uses every edge exactly once. So every edge is accounted for and there are no repeats. Thus every degree must be even.
How does the proof follow?
Edit: I figured it out.